Construction of Circles

Main Idea

Accurate construction of circles and regular figures depends on applying established geometric relationships involving equal radii, equal angular divisions, perpendicularity, angle bisectors, and perpendicular bisectors. Regular polygons may be inscribed in circles by joining equally spaced points on a circumference, or circumscribed about circles by drawing tangents at equally spaced points. In a regular polygon, the common centre determines both the inscribed circle, which touches every side, and the circumscribed circle, which passes through every vertex.

Key Concepts and Definitions

• Circle: A plane curve whose every point is at the same distance from a fixed point called the centre.
• Centre: The fixed point from which every point on the circle is equally distant.
• Radius: The distance from the centre of a circle to any point on its circumference.
• Diameter: A straight line passing through the centre and joining two points on the circumference; its length is twice the radius.
• Circumference: The complete boundary or curved line of a circle.
• Chord: A straight line segment joining any two points on a circle.
• Tangent: A straight line that touches a circle at exactly one point.
• Inscribed Circle: A circle drawn inside a polygon and touching each of its sides; it is also called the incircle.
• Circumscribed Circle: A circle drawn outside or around a polygon and passing through all its vertices; it is also called the circumcircle.
• Regular Polygon: A polygon having all sides equal and all interior angles equal.
• Apothem: The perpendicular distance from the centre of a regular polygon to any one of its sides; it is the radius of the inscribed circle.
• Circumradius: The distance from the centre of a regular polygon to any one of its vertices; it is the radius of the circumscribed circle.
• Concentric Circles: Two or more circles having the same centre but different radii.

Supporting Arguments and Evidence

• Accurate constructions require a sharp pencil, compass, ruler, set squares, and protractor where appropriate. Construction lines should be kept light, while final outlines should be drawn more darkly. Accuracy also depends on maintaining the compass setting, checking symmetry, and distinguishing construction lines from the final visible outline.

• To construct a circle of known radius, mark the centre, set the compass opening equal to the radius, and rotate the compass through one complete revolution. If a diameter is given, first locate its midpoint. This midpoint is the centre, and half the diameter is the radius.

• The circumference and area of a circle are calculated using

$$C=2\pi r=\pi d$$ and $$A=\pi r^2.$$ For numerical calculations, the approximate value of $\pi$ may be taken as $22/7$ or $3.14$.

• A regular polygon with $n$ sides is constructed by dividing the circle into equal central angles, each measuring

$$\frac{360^{\circ }}{n}.$$ A regular triangle is obtained by dividing a circle into three equal parts, a square into four equal parts, a regular pentagon into five parts, and a regular hexagon into six parts.

• To inscribe a regular polygon in a circle, the circumference is divided into equal angular parts, or equal chords are constructed. Consecutive points on the circumference are then joined. Joining points produces a regular polygon only when the points divide the circumference into equal arcs or equal central angles.

• For a regular hexagon inscribed in a circle, each side is equal to the radius of the circle because the six radii joining the centre to consecutive vertices form six equilateral triangles.

• To circumscribe a regular polygon about a circle, divide the circle into equal parts and draw tangents at the division points. Each tangent must be perpendicular to the corresponding radius at the point of contact. The intersections of consecutive tangents form the vertices of the circumscribed polygon.

• In a regular polygon, the centre is equidistant from all vertices and from all sides. The perpendicular distance from the centre to a side is the apothem and gives the radius of the inscribed circle. The distance from the centre to a vertex is the circumradius and gives the radius of the circumscribed circle.

• A perpendicular from the centre of a circle to a chord bisects the chord. This relationship supports accurate chord construction and the identification of symmetry in circular figures.

• The tangent at any point on a circle is perpendicular to the radius drawn to that point. This perpendicular relationship is essential when constructing a circumscribed regular polygon.

• To construct an inscribed circle in a regular polygon, bisect the angles or locate the polygon’s centre. Draw a perpendicular from the centre to any side; this perpendicular gives the radius of the incircle.

• To construct a circumscribed circle around a regular polygon, locate the intersection of the perpendicular bisectors of its sides. Use the distance from this centre to any vertex as the radius of the circumcircle.

• The inscribed and circumscribed circles of a regular polygon share one common centre. The perpendicular distance from that centre to a side determines the incircle, whereas the distance from the centre to a vertex determines the circumcircle.

• Circles with the same centre but different radii are concentric circles. This arrangement illustrates how a common centre can determine different circular boundaries.

What to Remember

Use equal radii, equal central angles, perpendicular lines, angle bisectors, and perpendicular bisectors as the basis of every construction. For a regular polygon, the apothem is the radius of the inscribed circle, while the circumradius is the radius of the circumscribed circle. Careful instrument use, constant compass settings, light construction lines, and accurate checking of symmetry are essential for reliable results.